Gibbs cluster measures on configuration spaces

TL;DR

This paper investigates Gibbs cluster measures on configuration spaces, establishing their quasi-invariance, deriving an integration-by-parts formula, and constructing associated stochastic dynamics, extending previous Poisson cluster measure results.

Contribution

It introduces a general framework for Gibbs cluster measures, proving quasi-invariance and Dirichlet form-based dynamics without requiring uniqueness of the background Gibbs measure.

Findings

Gibbs cluster measure $g_{cl}$ is quasi-invariant under compactly supported diffeomorphisms.

An integration-by-parts formula for $g_{cl}$ is established.

Associated stochastic dynamics are constructed via Dirichlet forms.

Abstract

The distribution $g_{cl}$ of a Gibbs cluster point process in $X=\mathbb{R}^{d}$ (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution $g$) is studied via the projection of an auxiliary Gibbs measure $\hat{g}$ in the space of configurations $\hat{gamma}=\{(x,\bar{y})\}\subset X\times\mathfrak{X}$, where $x\in X$ indicates a cluster "center" and $\bar{y}\in\mathfrak{X}:=\bigsqcup_{n} X^n$ represents a corresponding cluster relative to $x$. We show that the measure $g_{cl}$ is quasi-invariant with respect to the group $\mathrm{Diff}_{0}(X)$ of compactly supported diffeomorphisms of $X$, and prove an integration-by-parts formula for $g_{cl}$. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. These results are quite general; in particular, the uniqueness of the background Gibbs measure $g$ is not required. The paper is an extension of the earlier results for Poisson cluster measures %obtained by the authors [J. Funct. Analysis 256 (2009) 432-478], where a different projection construction was utilized specific to this "exactly soluble" case.